Given an input \(\mathbf {x}\)  such that \(\rho (\mathbf {x}) = c\) , a prime implicant explanation of \(\mathbf {x}\)  [1]}
(also referred to as a sufficient reason for \(\mathbf {x}\)  given \(\rho \)  [2]}) is a subset-minimal partial assignment \(\mathbf {x}^{\prime }\)  which is coherent with
\(\mathbf {x}\)  (i.e., \(\mathbf {x}\)  and \(\mathbf {x}^{\prime }\)  give the same values to the variables that are assigned in \(\mathbf {x}^{\prime }\) ) and which satisfies the property that for every extension \(\mathbf {x}^{\prime \prime }\)  of \(\mathbf {x}^{\prime }\)  over \(X_n\) , we have \(\rho (\mathbf {x}^{\prime \prime }) = c\) . The features assigned in \(\mathbf {x}^{\prime }\)  (and the way they are assigned) can be viewed as explaining why \(\mathbf {x}\)  has been classified by \(\rho \)  as a positive (or as a negative) instance. Formally:
